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July 14, 2012

The Greek experience, as we reconstitute it, accords special value to the “limit” and reemphasizes the long-recognize scandalousness of the irrational: the indecency of that which, in measurement, is immeasurable. (He who first discovered the incommensurability of the diagonal of the square perished; he drowned in a shipwreck, for he had met with a strange and utterly foreign death, the nonplace bounded by absent frontiers…)

The Writing of Disaster by Maurice Blanchot, tr. by Ann Smock, Lincoln: University of Nebraska Press, [1980]1995, p. 103. Part of this book is available via Google Books (although not the page where this excerpt appears).

Here’s the original French version:

C’est l’expérience grecque, telle que nous la reconstituons, qui privilégie la “limite” et qui confirme l’antique scandale de la rencontre de l’irrationnel, c’est-à-dire de la non-convenance de ce qui, dans la mesure, ne se mesure pas (le premier qui divulgua l’incommensurabilité de la diagonale du carré, périt, noyé dans un naufrage: c’est qu’il avait fait la rencontre d’une mort tout autre, le non-lieu du sans frontières…) (L’écriture du désastre, Paris: Gallimard, 1980, p. 160-161).

More information about the story surrounding the discovery of irrational numbers and the death of the man responsible for it is offered after the following comment.

• • •

In the speech he delivered on January 12, 2011 at the Together We Thrive: Tucson and America memorial to honor the victims of the 2011 Tucson shooting, President Barack Oabama made the following remark:

You see, when a tragedy like this strikes, it is part of our nature to demand explanations –- to try and pose some order on the chaos and make sense out of that which seems senseless.[…]

Scripture tells us that there is evil in the world, and that terrible things happen for reasons that defy human understanding. In the words of Job, “When I looked for light, then came darkness.” Bad things happen, and we have to guard against simple explanations in the aftermath. […] For the truth is none of us can know exactly what triggered this vicious attack. (The White House: “Remarks by the President at a Memorial Service for the Victims of the Shooting in Tucson, Arizona”, January 12, 2011).

After the shooting at the Dawson College in Montreal (Quebec, Canada) on September 13, 2006, Bloc Québécois Leader Gilles Duceppe told Radio-Canada:

“It’s tragic. We can never explain why these things happen” (CBC: “Politicians express sorrow over college shootings” September 14, 2006)

There are two apparently contradictory needs at work in each of those cases, as if we were standing precariously on a threshold. On one hand, as a collective, we need to somehow make sense of the senseless. What we experienced with each of those tragedies ―and indeed numerous similar ones― is the very limit of what can be thought, the limit of the thinkable itself. We came to live together out of fear of violent death and yet death came not from the outside, but from within. We need to find a way to process and share this experience, to make something out of it so it does not remain formless inside each of us and among us all.
On the other hand, there is this feeling that some things better stay hidden, shouldn’t be exposed. The crimes are so horrendous, surely the motives behind them are beyond the range of rational comprehension. Not only are there substantial difficulties in trying to catch hold of such motives, but more importantly some of us feel we shouldn’t even try in the first place. Like in the Greek legend, we feel there’s a risk in exposing ourselves to things that do not belong to the realm of order we know. To use Sir Thomas Little Heath’s own words (see below) such an understanding could come at a very high price: it could shake “the whole fabric” of our shared coexistence. And so we feel some aspects pertaining to those events are and should remain outside, alien to us, radically unknowable, unrepresentable and uncommunicable.

In 1976, Italian writer Primo Levi added an appendix to his book If This Is A Man (first published in 1947). The seventh part of this appendix has to do with the “fanatic hatred” expressed by the Nazis toward the Jews. Levi develops the idea that maybe there’s a moral imperative forbidding us to “comprehend” (literally “to catch hold of, seize”) this hatred for such a comprehension could equate or lead to a justification:

Perhaps one cannot—what is more, one must not—understand what happened, because to understand is almost to justify. Let me explain. Understanding a proposal or a form of human behavior means containing it, containing its author, putting oneself in his place, identifying with him. No normal human being will ever be able to identify with Hitler, Himmler, Goebbels, Eichmann, and the endless others. This dismays us, but at the same time it provides a sense of relief, because perhaps it is desirable that their words and their deeds cannot be comprehensible to us. (English tr. by Ruth Feldman, first published at The New Republic on February 17, 1986; French translation in Si c’est un homme, tr. by Martine Schruoffeneger, Paris: Pocket, p. 205).

The Pythagoreans were afraid the introduction of irrational numbers into the known order of things could destabilize what they knew about mathematics and therefore ―since mathematics were not an isolated discipline but a fundamental philosophy of life― could constitute a serious threat to the very foundation of their community:

Mais, en plus, ces philosophes mystiques pouvaient soupçonner, à juste titre, que le profane ne l’entendît pas ainsi et que, négligeant la domination du nombre sur l’incommensurable, il s’engouffrât dans la brèche de l’irrationalité. En conséquence, c’est toute la conception cosmo-théologique de la communion rationnelle des êtres, de la κοινωνια pythagoricienne dont Platon parle dans le Gorgias (508a), qui pouvait être mise à mal. (“La Découverte des incommensurables et le vetige de l’infini” by Jean-Luc Périllé, 2001, PDF)

Irrational numbers where said to be τῶν ἁλόγων πραγματείαν1: things or affairs which are not part of logos and cannot be talked about (ἄλογος also means “speechless”, “unutterable”, “unintelligible”: see Perseus). Primo Levi’s comment suggests something similar: in order to understand the Nazis, one would need to put himself or herself in their shoes and, in a way, would risk becoming as irrational, as inhuman as they were2. To understand their hatred is to risk introducing it among ourselves: it could spread like a contagion and destroy our humanity3.

And yet, our situation differs from the scandalous discovery made around 500 B.C. by the Pythagoreans (the account of this discovery is disputed, see below). For them precisely, it has to do with just that: a discovery. Whereas in modern times we are actually producing the incommensurable: it is immanent to our coexistence and our activities. We’re increasingly experiencing things which seem to be without limit, limitless.This is a common theme in the novels of American writer Don DeLillo. In Underworld (1997) here wrote:

Many things that were anchored to the balance of power and the balance of terror seem to be undone, unstuck. Things have no limits now. Money has no limits. I don’t understand money anymore. Money is undone. Violence is undone, violence is easier now, it’s uprooted, out of control, it has no measure anymore, it has no level of value. (New York: Scribner, [1997]2003, p. 76)

We are presented with human-produced events that we simply cannot grasp nor come to understand. They are beyond the scope of our capacity to accurately picture what they mean. Indeed it is hard, if not impossible, to imagine the systematic killing of millions of human beings during the Second World war. Rationality is somehow at a loss when it comes to the immediate effects produced by the explosion of an atomic bomb over Hiroshima, in the sunny morning of August 6, 1945. We were told about the Deepwater Horizon oil spill: an average of 53,000 barrels per day escaped in the ocean. We surely know the number, but how could we firmly grasp its meaning, its implications (scientific, ethical, political and so on)? Günther Anders called this phenomenon “the Promethean gap” (Das Prometheische Gefälle): the abysmal difference between what we can produce and our capacity to understand the actual effects of this production, between what we know and what we actually understand:

J’appelle “prométhéenne” la différence qui résulte du décalage entre notre “réussite prométhéenne” ―les produits fabriqués par nous, “fils de Prométhée”― et toutes nos autres performances, la différence qui existe une fois que nous avons réalisé que nous ne sommes pas à la hauteur du “Prométhée qui est en nous”. (L’Obsolescende de l’homme, tr. Christophe David, Paris: Encylopédie des Nuisances / Ivrea, [1956]2002, p. 301)

From this specific perspective, we can clearly see how the problem of our “togetherness” has also become in recent times a problem of epistemology. While the Greeks were afraid of the unknowable chaos lurking on the fringes of the cosmos, we have learned in modern times that it is not necessary to look that far away to be confronted with the incommensurable, with the very limits of knowledge. Modern times expose us to experiences which often do not share anything common ―no common unit of measurement4― with anything previously known. This situation, as Walter Benjamin once wrote, has “diminished the communicability of experience” (“The Storyteller” in Illuminations, tr. by Harry Zohn, New York: Schocken Books, p. 93). And thus the (epistemological) problem of understanding what’s happening to us finds its echo in the (political) problem of living together while sharing a common experience of the world.

This precarious situation ―the “apparent contradiction” I opened this entry with: to know what is or must remain unknowable― could be said to be the immediate horizon of our coming coexistence. The problem could thus be formulated that way: how are we going to create adequate conditions for our togetherness while facing ―if not producing― the very threat which could dissolve it?5

Related entries:

• • •

The Greek legend quoted by Maurice Blanchot at the very beginning of this entry is both well known and clouded in various controversies6. It has been recounted in multiple forms and attributed to various authors. Below one can find complementary material related to this legend. Both first and second hand sources as well as modern analysis and references are prodivided. It really doesn’t answer all the questions, but it helps make sense of some of them.
First, it’s important to know there are numerous litigious aspects to this legend:

  • There are disputes about the discovery of “the doctrine of incommensurables” and thus the theory of irrationals. Some say it was discovered by Pythagoras (c. 570- c.495) himself, other by Pythagoreans in general. In some instance one man is singled out: Hippasus (5th century BC), but this is also challenged. Finally, it is often suggested that the mathematicians in India were already aware of those concepts. As usual, a simple principle applies: if the tale is not backed by actual sources, it should be considered the personal tale of the author telling it.

  • There are also discussions about how the discoverer was killed (thrown overboard or drown in a shipwreck) or even if he was really killed in the first place.

  • Often, this legend is either attributed to Pappus of Alexandria (c. 290-c. 350) or to Proclus (412-485). Both wrote about it in the comments or scholia they produced regarding Euclid’s famous Elements

  • Finally, nothing is sure about the idea that the discovery of incommensurability comes from the study of the square’s diagonal.

One of the earliest account of the legend comes from Iamblichus (c.245–c.325) who wrote about it in his Life of Pythagoras (or Pythagoric Life). An English translation (from the Greek original) by Thomas Taylor is available online (London: J.M. Watkins, 1818, Internet Archive). Iamblichus points to two different aspects of the legend: 1) Hippasus is credited for the discovery, but really it was Pythagoras himself who made it in the first place; 2) The discovery has to do with pentagons, not square roots. See specifically pp. 47-48:

With respect to Hippasus however especially, they assert that he was one of the Pythagoreans, but that in consequence of having divulged and described the method of forming a sphere from twelve pentagons, he perished in the sea, as an impious person, but obtained the renown of having made the discovery. In reality, however, this as well as every thing else pertaining to geometry, was the invention of that man; for thus without mentioning his name, they denominate Pythagoras.

The original Greek text for Iamblichus’s account is reproduced in Volume 1 of Ivor Thomas’s Selections Illustrating the History of Greek Mathematics which is also available online (Cambridge: Harvard University Press, [1939]1957, pp. 222, 224; Internet Archive).

In a footnote pertaining to the shipwreck mentioned by Iamblichus, the English translator of his Life of Pythagoras added:

The same thing is said by the Pythagoreans to have befallen the person who first divulged the theory of incommensurable quantities. See the first scholium on the 10th book of Euclid’s Elements, in Commandine’s edition, fol. 1572.

A scholium (or scholia in the plural form) is a commentary made about a text. Scholia to Euclid’s Elements (which counts thirteen books) have been either added to each of Euclid’s books (in the margins for example), or published separately. For more about the history of scholia added to Euclid’s Elements see “The Scholia” in chapter VI of Introduction and Books 1,2 by Euclid, Volume 1 of The Thirteen Books of Euclid’s Elements, translated with introduction by Sir Thomas Little Heath (republication of the Second Edition Unabridged, New York: Dover, 1956, p. 64).

When mentioning the Greek legend, those scholia are either attributed to Pappus of Alexandria or to Proclus. In fact, the very same sentences are sometimes attributed to two different authors. There may be a simple explanation to this confusion. In the first volume of his translation of Euclid’s Elements (the one I just referred to), Sir Thomas Little Heath explains that the scholia from Poppus ―and especially those concerning the tenth book of Euclid’s Elements― were included in Proclus’s own scholia:

The rest of Schol. Vat. (on Books II.-XIII.) are essentially of the same character as those on Book I., containing prolegomena, remarks on the object of the propositions, critical remarks on the text, converses, lemmas; they are in general, exact and true to tradition. The reason of the resemblance between them and Proclus appears to be due to the fact that they have their origin in the commentary of Pappus, of which we know that Proclus also made use. In support of the view that Pappus is the source, Heidberg paces some of the Schol. Vat. to Book X. side by side with passages from the commentary of Pappus in the Arabic translation discovered by Woepcke (…) (Introduction and Books 1,2 by Euclid, Volume 1 of The Thirteen Books of Euclid’s Elements, translated with introduction by Sir Thomas Little Heath, Courier Dover Publications, 1956, 66; see also pp. 24-25)

Τhe original Greek text of Proclus’s Commentary on the First Book of Euclid’s Elements is available online over at the Internet Archive: Procli Diadochi in primum Euclidis Elementorum librum commentarii edited and introduced by Gottfried Friedlein (1873). A partial preview of the English translation by Glenn Raymond Morrow is also available on Google Books: A Commentary on the First Book of Euclid’s Elements, Princeton: Princeton University Press, 1992). Excerpts relevant in the case at hand here are quoted by Sir Thomas Little Heath in the introduction to his translation of Book X of Euclid’s Elements:

The scholium quotes further the legend according to which “the first of the Pythagoreans who made public the investigation of these matters [the doctrine of incommensurables] perished in a shipwreck,” conjecturing that the authors of this story “perhaps spoke allegorically, hinting that everything irrational and formless is properly concealed and, if any soul should rashly invade this region of life and lay it open, it would be carried away into the sea of becoming and overwhelmed by its unresting currents.” There would be a reason also for keeping the discovery of irrationals secret for the time in the fact that it rendered unstable so much of the groundwork of geometry as the Pythagoreans had based upon the imperfect theory of proportions which applied only to numbers. We have already, after Tannery, referred to the probability that the discovery of incommensurability must have necessitated a great recasting of the whole fabric of elementary geometry, pending the discovery of the general theory of proportion applicable to incommensurable as well as to commensurable magnitudes. (Books X-XIII and Appendix by Euclid, Volume 3 of The Thirteen Books of Euclid’s Elements, translated with introduction by Sir Thomas Little Heath, Second Edition Unabridged, Cambridge: University Press, 1908, p. 1; a different version from 1956 by Dover is available on Google Books.)

It is possible to compare this source to the original comments made by Pappus of Alexandria. He did indeed explicitly mentioned the Greek legend in his Commentary on Book X of Euclid’s Elements. The text was translated to English from the original Arabic version (both are available online):

Since this treatise (i.e. Book X of Euclid) has the aforesaid aim and object, it will not be unprofitable for us to consolidate the good which it contains. Indeed the sect (or school) of Pythagoras was so affected by its reverence for these things that a saying became current in it, namely, that he who first disclosed the knowledge of surds or irrationals and spread it abroad among the common herd, perished by drowning: which is most probably a parable by which they sought to express their conviction that firstly, it is better to conceal (or veil) every surd, or irrational, or inconceivable in the universe, and, secondly, that the soul which by error or heedlessness discovers or reveals anything of this nature which is in it or in this world, wander [thereafter] hither and thither on the sea of non-identity (i. e. lacking all similarity of quality or accident), immersed in the stream of the coming-to-be and the passing away, where there is no standard of measurement. (see Pappus’s Commentary: Arabic Text and English Translation by William Thomson and Gustav Jung, Harvard Semitic series vol. 8, Cambridge: Harvard University Press, 1930, Part 1, §2 p. 64; PDF)

Last but not least, Richard Crew, Professor for the Department of Mathematics at the University of Florida wrote a very detailed and well documented account of the legend challenging its “allegorical” foundation: see Pythagoras, Hippasos, and the Square Root of Two, August 2000. Crew underlines an important aspect of the legend, the very idea that Pythagoreans discovered the irrational numbers. He does so from the perspective of Sir Thomas Heath’s lexical observations:

There is a passage from Proclus’s Commentary on the First Book of Euclid’s Elements which, in one version, is translated by Heath as asserting that Pythagoras discovered “the theory of irrationals” (ten ton alogon pragmateian). There is, however, a textual problem here, in that the word alogon, “irrational,” appears in other manuscripts as analogon, in which case the assertion is that Pythagoras discovered the “theory of proportions.” Heath remarks that the form ton analogon is not correct Greek; the right reading may be either ton analogion (“proportions”) or ton ana logon (“proportionals”), and he opts for the latter. This reading seems to be the generally accepted one now, and is the one used in the translations of Proclus by Thomas, and by Morrow. All of these authors point out that it is quite unlikely that anyone as early as Pythagoras could have had a general “theory of irrationals”; at most, he or his disciples could have discovered the irrationality of the square root of 2, and possibly a few other ratios. One knows, in any case, how far the study had progressed by the time of the generation of Plato, for Theaetetus poses, in the dialogue named after him, precisely the problem of creating such a theory.

He goes on to conclude his short but penetrating essay on this matter by suggesting:

If this account is true – and Heath makes a plausible case for it – then the discovery of irrational ratios was not so much of a philosophical or metaphysical crisis as a purely mathematical one. This might be construed as a disappointment for the philosphers, but not for mathematicians.

• • •

1. For a commentary on this locution see Professor Richard Crew’s comment at the very end of this entry. For more references and discussion see also The Mathematics of Plato’s Academy. A New Reconstruction by David H. Fowler, Oxford: Oxford University Press, 1999, 8.3(b)iv, pp. 292-293. The expression is used by Proclus in his Commentary on the First Book of Euclid’s Elements. Below is a screenshot of Friedlein’s edition:

“τῶν ἁλόγων πραγματείαν” in Proclus’s Commentary on Euclid’s Element (see Procli Diadochi in primum Euclidis Elementorum librum commentarii, edited and introduced by Gottfried Friedlein, 1873, p. 65)

See online Procli Diadochi in primum Euclidis Elementorum librum commentarii edited and introduced by Gottfried Friedlein, 1873, p. 65. ↩︎︎

2. Hannah Arendt’s Eichmann in Jerusalem (1963) could be cited as an interesting counter-argument to Primo Levi’s moral imperative. She attended to Eichmann’s trial in Jerusalem in 1961 precisely to confront the man, listen to him and try to understand. She famously (and controversially) found no trace of monstrosity but instead developed her thesis about the “banality of evil”. ↩︎︎

3. Roberto Esposito makes a similar remark in his book Communitas:

Whereas knowledge tends to stitch up every tear, non-knowledge consists in holding open the opening that we already are [nel tenere aperta I apertura che gia siamo]; of not blocking but rather of displaying the wound in and of but existence. (tr. by Thimothy Campbell, Stanford: Stanford University Press, [1998]2010, p. 119).

The important thing to keep in mind here is that this opening is not necessarily bad: it comes with high risks, but also with opportunities. From this point of view, the unwillingness to risk anything is a much greater threat than non-knowledge. ↩︎︎

4. With Bataille, Blanchot Nancy and Esposito the argument put forward is even stronger: what we have in common is that we have nothing in common, we do not share any common unit of measurement. Maurice Blanchot explicitly addresses this issue in his book The Infinite Conversation:

But it may be that I connot give the measure of equality its true sense unless I maintain the absence of common measure that is my relation to autrui. (tr. by Susan Hanson, Minneapolis: University of Minnesota Press, [1969] 1993, p. 64; originally published as L’Entretien ifini, Paris:Gallimard, 1969, p. 92)

See also Georges Bataille in Oeuvres Complètes:

(…) la disproportion, l’absence de commune mesure entre diverses entités humaines étant en quelque sorte un des aspects de la disproportion générale entre l’homme et la nature. (Tome 1: “Premiers écrits, 1922-1940”, Paris: Gallimard, 1970, p. 182) ↩︎︎

5. The threshold represents the higher necessity as well as the higher risk. I wrote about that following an similar comment by Adorno and Horkheimer: see Classification by Adorno and Horkheimer (1947).

It would be tempting to suggest that we somehow need to find a way to reappropriate the communicability modern experiences. However, it could be argued that communicability is precisely what has been usurped from us: it has already been appropriated, that is made proper and thus not common anymore. Following this line of thought, it could interesting to explore the potential of collective expropriation as a mean to fully share again our individual experiences: what (critical) potential could be found (or thought) regarding “inappropriability”, the quality of being inappropriable? Even or especially if this means to re-think what is meant by “communicability” and “incommunicability”, or to make present, between us all, something irrepresentable.

On the appropriation of communicability, see Parables for the virtual by Brian Massumi which I quoted before: “Because Montréal, this city, my city, is burning with life, but not burning at all.”. On “inappropriability” see “Érudition et fétichisme” by Éric Méchoulan (in La Littérature en puissance. Autour de Giorgio Agamben, 2006) and “Is Everything Political” by Jean-Luc Nancy (in The Truth of Democracy, 2010). Following Bataille and Blanchot, Roberto Esposito writes about the “inappropriability of death” in Communitas ([1998]2010). Finally on the concept of “communicability” see “Communication as Communicability” by Briankle G. Chang (in Communication as… Perspectives on Theory, 2006). ↩︎︎

6. [UPDATE – July 29, 2012] I’m aware of the piece American filmmaker Erroll Morris wrote about this very legend for The New York Times in 2011 (“The Ashtray: Hippasus of Metapontum (Part 3)”, March 8, 2011). It’s a peculiar piece in many ways. It’s the third part of his five parts series against Thomas Kuhn’s “incommensurability” (I wrote about this series before: see here and here). Although I greatly appreciate Errol Morris’s work as a filmmaker and as a writer in his previous series about the nature of images, I thought (and still think) his piece on Kuhn was weak. The third part which directly addresses the Greek legend about the discovery of incommensurable numbers doesn’t help to shake this impression, on the contrary. Morris seems more eager to hold to his belief that Kuhn was wrong than to produce a rigorous investigation.

Morris’s main argument is that there was no crisis at all ―philosophical nor mathematical― to begin with. The problem is that in Morris turns controversial arguments into facts to serve his attempt at debunking Kuhn’s theory (both of incommensurability and, therefore, of “paradigm shift”). For example, Morris states:

Someone believed that there should have been a crisis even if there wasn’t any. They believed that the Pythagoreans should have been upset about the discovery of incommensurable magnitudes. But it was a retrospective belief, that is, a belief formed hundreds, if not thousands of years, after the crisis was supposed to have occurred. [emphasis in the original text]

In order to suggest that the belief in a crisis is a modern invention, one has to completely ignore the writings of both Pappus of Alexandria and Proclus. Indeed Morris merely quotes a small portion of Pappus’s comment and, more surprisingly coming from him, doesn’t provide a source for the text he uses. Yet, both the original text and its English translation are easily found online (see above for those references). Instead, Morris relies on modern interpretation of those texts (mostly from Wilbur Knorr and David H. Fowler).

My argument is not, contra Morris, that there was indeed a crisis: I don’t know that to be a fact. What I know is that there are ancient texts (at least three: Iamblichus, Pappus and Proclus) mentioning a “legend”, an “allegory” or a “saying” (which were already at the time interpreted by Pappus and Proclus as such: legend, allegory, saying). I also know that controversies have surrounded and still surround the interpretation of those texts.↩︎︎

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